28 Mart 2016 Pazartesi

Left as an exercise: the joke of scientific writing

We all have been students and we all have suffered a certain attitude called left as an exercise or Proof: Trivial. QED from textbook writers. If we are to name names, Thomas W. Hungerford, with his Algebra is number one on my list. That book is the super massive black hole of dense writing. Perhaps Geroch, Dirac or Abraham and Marsden come second. Below is an historical example, or rather a proof, that some authors leave these exercises without actually solving them. (No, I do not claim that TWH is as such.) It is about constructing a triangle whose angle-bisectors are given with ruler and compass, which has no solution other than special cases. This inverse geometric problem has been the subject of a PhD thesis by Richard Philip Baker as late as 1903 and his solution uses techniques of algebraic geometry. I reproduce page V of the thesis here. The first five paragraphs clearly state the difficulty of the problem, whereas the final one refers to the Proof: Trivial. attitude.

THE HISTORY OF THE PROBLEM

The problem of constructing a triangle when the lengths of the bisectors of the angles are given has been an outstanding problem among geometers from the time of Pascal and certainly from the time of Euler.

Brocard has summed up the literature, dealing almost entirely with special cases, of which the most extensive treatment appears to be the solution of the problem when one angle is a right angle due to Marcus Baker. This problem is of the sixth order. Among many special treatments that appear in the smaller journals the fundamental problem of determining the character of the algebraic irrationality involved is not mentioned. As a result apparently conflicting statements occur as to the order of the equation concerned, this depending on accidental choice of the parameter field.

The only paper dealing with the general case where the internal bisector formulas are used is P. Barbarin's. The case where the external formulas are to be used and the case where two of the assigned bisectors refer to the same vertex is not treated in general in any paper known to the writer.

Barbarin proved that the general internal problem could be solved by the solution of an algebraic equation of order not greater than twelve. The irreducibility and group of the equation are not discussed, and as the equation itself is not set out explicitly further reduction of the order of the problem is not precluded.

The method of attack used by Barbarin is to solve first the problem when an angle and two bisectors are given, and to use the result as a basis for attacking the general problem. The necessary sacrifice of symmetry prevents any explicit comparison with the solution given in this paper except at the cost of labor disproportionate to the result.

The problem must have an extensive domestic history in the schools: Barbarin charges that E. Catalan was among those who have proposed it as an elementary exercise, and from a Russian scholar the writer learns that it has been there extensively used in the schools as a standard set-back for ambitious young geometers.

R. P. Baker was a lawyer and practiced law in Texas. He later did a PhD at the University of Chicago in the field of algebraic geometry. (Another lawyer turned into scientist at the University of Chicago is Edwin Hubble, the man who discovered that our universe is expanding.) Baker is also known for producing and selling three dimensional models of geometrical objects. Some of these items are now museum pieces and displayed at various math departments in the US.